Effects of dispersion parity-violating interaction in electron scattering and atoms

Exchange of two neutrinos (as well as other fermions) generates a long-range parity-violating (PV) potential of the form $\sim G^2/r^5$, with characteristic range $\hbar/(2m_\nu c)$. In atomic systems the corresponding matrix elements converge at distances $r < 10/M_Z$, so that the interaction between electron and quarks effectively reduces to a contact term $\sim G^2 M_Z^2 \delta^{3}(\vec{r}) \sim G\,\alpha\,\delta^{3}(\vec{r})$. This interaction produces a $-0.8\%$ correction to the effective weak charge of cesium, resolving the $2\sigma$ discrepancy between the Standard Model prediction and the measured Cs parity-violation amplitude. The corresponding value of the weak mixing angle is $\sin^2\theta_W = 0.2375(19)$ at $q^2 \approx 0$, in agreement with the Standard Model prediction $\sin^2\theta_W = 0.23873$. The relative correction to the proton weak charge is about $3\%$, with a similar correction for the electron weak charge. Using these results, we revisit the limits on an additional $Z'$ boson and obtain a constraint on isospin-conserving oblique radiative corrections characterized by the Peskin--Takeuchi parameter $S = -0.32(53)$, at $q^2 \approx 0$.